Statement-wise Verification

• Statement 1 - Correct.
  The divergence of a vector field represents the net outward flux per unit volume from a point and indicates the presence of sources or sinks.
  Divergence is a scalar measuring net outward flux density/source strength at a point.
• Statement 2 - Correct.
  The curl of a vector field measures the rotational tendency of the field and is zero for conservative fields.
  Curl is a vector measuring local rotational tendency; for a sufficiently smooth conservative field F=−∇V, ∇×F=0.
• Statement 3 - Incorrect.
  The gradient of a scalar field always results in a scalar quantity representing rate of change.
  ∇φ is a vector field pointing in the direction of steepest increase of scalar φ.
• Statement 4 - Correct.
  Divergence theorem relates surface integral of a vector field over a closed surface to volume integral of divergence.
  The divergence theorem converts closed-surface flux into a volume integral of divergence.

Core Concept

The three basic vector differential operators have distinct outputs: gradient maps scalar→vector, divergence maps vector→scalar, and curl maps vector→vector. Remembering input/output type is often enough to eliminate distractors.

Formula / Key Relationship

grad φ=∇φ; div A=∇·A; curl A=∇×A; ∮_S A·dS = ∫_V (∇·A)dV.

Step-by-Step Check

Type-checking immediately disproves statement 3. The geometric meanings and divergence theorem confirm 1, 2 and 4, yielding A.

Why the Other Options Are Wrong

Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 2, 4. Option C is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4. Option D is incorrect because it includes incorrect statement(s) 3.

Answer-Key Verification Note

The uploaded provisional key marks C . After independent technical verification, the defensible answer is A . This discrepancy is stated explicitly rather than silently copying the provisional key.

Exam Shortcut / Approach

Use the mnemonic: Grad grows a scalar into a vector; Div gives a scalar; Curl gives a vector.

Common Mistake

Calling gradient a scalar because the original field is scalar.

Quick Revision

Vector Differential Operators is linked with Gradient Divergence and Curl, Vector Calculus and Electromagnetic Fields. Use the mnemonic: Grad grows a scalar into a vector; Div gives a scalar; Curl gives a vector.

Statement-wise Verification

• Statement 1 - Correct.
  The divergence of a vector field represents the net outward flux per unit volume from a point and indicates the presence of sources or sinks.
  Divergence is a scalar measuring net outward flux density/source strength at a point.
• Statement 2 - Correct.
  The curl of a vector field measures the rotational tendency of the field and is zero for conservative fields.
  Curl is a vector measuring local rotational tendency; for a sufficiently smooth conservative field F=−∇V, ∇×F=0.
• Statement 3 - Incorrect.
  The gradient of a scalar field always results in a scalar quantity representing rate of change.
  ∇φ is a vector field pointing in the direction of steepest increase of scalar φ.
• Statement 4 - Correct.
  Divergence theorem relates surface integral of a vector field over a closed surface to volume integral of divergence.
  The divergence theorem converts closed-surface flux into a volume integral of divergence.

Core Concept

The three basic vector differential operators have distinct outputs: gradient maps scalar→vector, divergence maps vector→scalar, and curl maps vector→vector. Remembering input/output type is often enough to eliminate distractors.

Formula / Key Relationship

grad φ=∇φ; div A=∇·A; curl A=∇×A; ∮_S A·dS = ∫_V (∇·A)dV.

Step-by-Step Check

Type-checking immediately disproves statement 3. The geometric meanings and divergence theorem confirm 1, 2 and 4, yielding A.

Why the Other Options Are Wrong

Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 2, 4. Option C is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4. Option D is incorrect because it includes incorrect statement(s) 3.

Answer-Key Verification Note

The uploaded provisional key marks C . After independent technical verification, the defensible answer is A . This discrepancy is stated explicitly rather than silently copying the provisional key.

Exam Shortcut / Approach

Use the mnemonic: Grad grows a scalar into a vector; Div gives a scalar; Curl gives a vector.

Common Mistake

Calling gradient a scalar because the original field is scalar.

Quick Revision

Vector Differential Operators is linked with Gradient Divergence and Curl, Vector Calculus and Electromagnetic Fields. Use the mnemonic: Grad grows a scalar into a vector; Div gives a scalar; Curl gives a vector.

Statement-wise Verification

• Statement 1 - Correct.
  The divergence of a vector field represents the net outward flux per unit volume from a point and indicates the presence of sources or sinks.
  Divergence is a scalar measuring net outward flux density/source strength at a point.
• Statement 2 - Correct.
  The curl of a vector field measures the rotational tendency of the field and is zero for conservative fields.
  Curl is a vector measuring local rotational tendency; for a sufficiently smooth conservative field F=−∇V, ∇×F=0.
• Statement 3 - Incorrect.
  The gradient of a scalar field always results in a scalar quantity representing rate of change.
  ∇φ is a vector field pointing in the direction of steepest increase of scalar φ.
• Statement 4 - Correct.
  Divergence theorem relates surface integral of a vector field over a closed surface to volume integral of divergence.
  The divergence theorem converts closed-surface flux into a volume integral of divergence.

Core Concept

The three basic vector differential operators have distinct outputs: gradient maps scalar→vector, divergence maps vector→scalar, and curl maps vector→vector. Remembering input/output type is often enough to eliminate distractors.

Formula / Key Relationship

grad φ=∇φ; div A=∇·A; curl A=∇×A; ∮_S A·dS = ∫_V (∇·A)dV.

Step-by-Step Check

Type-checking immediately disproves statement 3. The geometric meanings and divergence theorem confirm 1, 2 and 4, yielding A.

Why the Other Options Are Wrong

Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 2, 4. Option C is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4. Option D is incorrect because it includes incorrect statement(s) 3.

Answer-Key Verification Note

The uploaded provisional key marks C . After independent technical verification, the defensible answer is A . This discrepancy is stated explicitly rather than silently copying the provisional key.

Exam Shortcut / Approach

Use the mnemonic: Grad grows a scalar into a vector; Div gives a scalar; Curl gives a vector.

Common Mistake

Calling gradient a scalar because the original field is scalar.

Quick Revision

Vector Differential Operators is linked with Gradient Divergence and Curl, Vector Calculus and Electromagnetic Fields. Use the mnemonic: Grad grows a scalar into a vector; Div gives a scalar; Curl gives a vector.